Finite Group Databasebeta

← #31779337

Subgroups — #31779337

Order 17536.

252 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22577953C211yesmaximal, characteristic
2#22577271C211yesmaximal, characteristic, complemented
3#22577138C214yesmaximal, complemented
4#432×C137C211yesmaximal, characteristic, complemented
5C2×#14555510—21no—
6#17984088C411yescharacteristic, complemented
7#17984085C411yescharacteristic, complemented
8#17984076—21no—
9#17984062—21no—
10#17984062C2212yes—
11C4×D548—21no—
12C4×D548C412yescomplemented
13#17983995—21no—
14#17983995—21no—
15#17983995C2212yes—
16C2×C42×C137C2211yescharacteristic
17C4⋊C8×C137—21no—
18C22⋊C8×C137C2212yes—
19#97×C137—21no—
20#14555510—44no—
21C2×Dic274—21no—
22C2×D548—21no—
23C2×D548—21no—
24C2×D548C2×C412yes—
25C2×C22⋊D137—22no—
26C22×Dic137—22no—
27C2×C4×D137—22no—
28C4⋊Dic137—21no—
29C4⋊Dic137C2×C412yes—
30#14555487—22no—
31#14555487—22no—
32C22×C4×C137D411yescharacteristic
33C22×C4×C137D411yescharacteristic
34C22×C4×C137C2311yescharacteristic
35C2×C8×C137—22no—
36C42×C137C2×C411yescharacteristic
37C42×C137C2×C411yescharacteristic
38C42×C137D411yescharacteristic
39C42×C137D411yescharacteristic
40C22×D137—22no—
41Dic274—42no—
42D548—42no—
43D548—42no—
44D548—42no—
45C22⋊D137—44no—
46C2×Dic137—22no—
47C2×Dic137—42no—
48C2×Dic137—42no—
49C2×Dic137—42no—
50C4×D137—44no—
51C23×C137C2×D411yescharacteristic
52C2×C4×C137—22no—
53C2×C4×C137—41no—
54C2×C4×C137—41no—
55C2×C4×C137C22⋊C411yescharacteristic
56C2×C4×C137C22⋊C411yescharacteristic
57C2×C4×C137C22⋊C411yescharacteristic
58C2×C4×C137C22⋊C411yescharacteristic
59C2×C4×C137C22×C411yescharacteristic
60C2×C4×C137C2×D411yescharacteristic
61C8×C137—42no—
62C22×C137—21no—
63C22×C137—21no—
64C22×C137C2×C22⋊C411yescharacteristic
65C22×C137AGL(1,ℤ/8)12yes—
66C2×D137—42no—
67C2×D137—42no—
68C2×D137—42no—
69C4×C137—21no—
70C4×C137—21no—
71C4×C137—21no—
72C4×C137—21no—
73C4×C137—21no—
74C4×C137—21no—
75Dic137—42no—
76Dic137—82no—
77C2×C137—22no—
78C2×C137#41011yescharacteristic
79C2×C137#41811yescharacteristic
80C2×C137#42111yescharacteristic
81D137—82no—
82C137#151411yescharacteristic, complemented, Sylow
83#1514—1371nomaximal, Sylow
84#532—1371no—
85#432D13711yescharacteristic, complemented
86#383—1371no—
87#327—1374no—
88C2×#41—2741no—
89C4⋊Q8—1371no—
90#127—1371no—
91#124—2741no—
92#121—2741no—
93#121—1372no—
94C4×D4—2741no—
95C4×D4—1372no—
96C2×C42C2×D13711yescharacteristic
97C4⋊C8—21no—
98C22⋊C8C2×D13712yes—
99#97—21no—
100#95—2741no—
101#95—2741no—
102#95—1372no—
103#41—5484no—
104C2×Q8—2741no—
105C2×D4—2742no—
106C2×D4—2741no—
107C2×D4—2741no—
108C2×D4—1372no—
109C22×C4—2742no—
110C22×C4—2742no—
111C22×C4D54811yescharacteristic
112C22×C4D54811yescharacteristic
113C22×C4C22×D13711yescharacteristic
114C2×C8—22no—
115C4⋊C4—2741no—
116C4⋊C4—1372no—
117C22⋊C4—2742no—
118C22⋊C4—2742no—
119C42C4×D13711yescharacteristic
120C42C4×D13711yescharacteristic
121C42C22⋊D13711yescharacteristic
122C42C22⋊D13711yescharacteristic
123C23—2742no—
124C23C2×D54811yescharacteristic
125Q8—5482no—
126D4—5482no—
127D4—5482no—
128D4—5482no—
129D4—5484no—
130C2×C4—22no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—2742no—
134C2×C4—5484no—
135C2×C4—5482no—
136C2×C4—5482no—
137C2×C4—5482no—
138C2×C4#1455548711yescharacteristic
139C2×C4#1455548711yescharacteristic
140C2×C4#1455548711yescharacteristic
141C2×C4#1455548711yescharacteristic
142C2×C4C2×C4×D13711yescharacteristic
143C2×C4C2×C22⋊D13711yescharacteristic
144C8—42no—
145C22—21no—
146C22—21no—
147C22—5482no—
148C22—5482no—
149C22—5482no—
150C22C2×#1455548711yescharacteristic
151C22#1798411612yes—
152C4—21no—
153C4—21no—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—5482no—
159C4—10962no—
160C2—22no—
161C2—10962no—
162C2#2257737611yescharacteristic
163C2#2257740911yescharacteristic
164C2#2257741911yescharacteristic
subgroupG/H
centerC22C2×#14555487
commutatorC2×C4×C137C22×C4
FrattiniC22×C4C22×D137
Fitting#432×C137C2
socleC22×C137C2×C22⋊C4
prime radical#22577953C2
cosocleC22×C4×C137C23
hypercenter#432D137
hypocenterC137#1514
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22577271C2semidirect
3. #22577138C2semidirect
4. #432×C137C2semidirect
6. #17984088C4semidirect
7. #17984085C4semidirect
12. C4×D548C4semidirect
82. C137#1514semidirect
85. #432D137semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225779532
#225772712
#225771382
#432×C1372
#1514137